These examples illustrate that simple solutions form attractive areas in the optimization landscape: once a PINN finds a simple, locally consistent solution in some areas (typically far away from the initial conditions), it may be very difficult to change it. This leads to a final solution which is a combination of the correct solution and a wrong one.
The ability to escape from wrong solutions largely depends on the optimizer used for training PINNs. Work [1]} reports weak results of PINNs when using the LBFGS optimizer. We however obtain much better results for the same systems using the Adam optimizer [2]} (see Tables REF , REF , REF ). Note, that in the final results for the method trained with Adam we included other optimization tricks such as normalizing the PINN inputs to \([-1, 1]\)  that typically helps to improve the accuracy. Additionally, using the initial condition points to calculate the PDE loss (REF ) typically improves the results.
